find the image of the circle mod z=1 by transformation w=z+2+4i

To find the image of the circle under the transformation \( w = z + 2 + 4i \), we follow these steps:

### 1. Equation of the Circle:
The given equation is \( |z| = 1 \), which represents a circle with radius 1 centered at the origin in the complex plane.

### 2. Transformation:
The transformation \( w = z + 2 + 4i \) can be interpreted as a translation of the complex number \( z \) by the vector \( 2 + 4i \). This will move every point on the circle by that amount.

### 3. Apply the Transformation:
Since \( w = z + 2 + 4i \), the transformation simply shifts the center of the circle from the origin to the point \( 2 + 4i \). Therefore, the radius remains the same, but the center is now at \( (2, 4) \).

### 4. Result:
The image of the circle \( |z| = 1 \) under the transformation \( w = z + 2 + 4i \) is another circle with:
- **Center**: \( (2, 4) \)
- **Radius**: \( 1 \)

### Final Equation of the Circle:
The equation of the new circle in the complex plane is \( |w - (2 + 4i)| = 1 \), which represents the image of the original circle after the transformation.

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Here are five related questions to explore further:
1. What is the geometric meaning of a Möbius transformation?
2. How do you find the image of a line under complex transformations?
3. What happens to circles under general linear transformations?
4. Can we generalize this transformation to \( w = az + b \)?
5. How do transformations affect the curvature of objects?

**Tip:** Translations in the complex plane are simple shifts of all points by a fixed vector. They do not change the size or shape of geometric objects like circles.

Explore how the complex transformation w = z + 2 + 4i shifts the circle |z| = 1 in the complex plane. Learn about geometric transformations, translations, and how the center and radius of a circle are affected. Suitable for advanced high school students.

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